3.1455 \(\int \frac{1}{x^3 \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=203 \[ \frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^
(5/4)) - (b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(5/4))
- (b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]
*a^(5/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(
8*Sqrt[2]*a^(5/4))

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Rubi [A]  time = 0.383974, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^8)),x]

[Out]

-1/(2*a*x^2) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^
(5/4)) - (b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(5/4))
- (b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]
*a^(5/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(
8*Sqrt[2]*a^(5/4))

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Rubi in Sympy [A]  time = 61.8301, size = 187, normalized size = 0.92 \[ - \frac{1}{2 a x^{2}} - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 a^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 a^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**8+a),x)

[Out]

-1/(2*a*x**2) - sqrt(2)*b**(1/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*x**2 + sqrt(a) +
sqrt(b)*x**4)/(16*a**(5/4)) + sqrt(2)*b**(1/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*x*
*2 + sqrt(a) + sqrt(b)*x**4)/(16*a**(5/4)) + sqrt(2)*b**(1/4)*atan(1 - sqrt(2)*b
**(1/4)*x**2/a**(1/4))/(8*a**(5/4)) - sqrt(2)*b**(1/4)*atan(1 + sqrt(2)*b**(1/4)
*x**2/a**(1/4))/(8*a**(5/4))

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Mathematica [A]  time = 0.256655, size = 385, normalized size = 1.9 \[ \frac{\sqrt{2} \sqrt [4]{b} x^2 \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt{2} \sqrt [4]{b} x^2 \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt{2} \sqrt [4]{b} x^2 \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt{2} \sqrt [4]{b} x^2 \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )-8 \sqrt [4]{a}}{16 a^{5/4} x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^8)),x]

[Out]

(-8*a^(1/4) + 2*Sqrt[2]*b^(1/4)*x^2*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(
1/8)] + 2*Sqrt[2]*b^(1/4)*x^2*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]
- 2*Sqrt[2]*b^(1/4)*x^2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]] + 2*Sq
rt[2]*b^(1/4)*x^2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] - Sqrt[2]*b^
(1/4)*x^2*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - Sqrt[2]*b
^(1/4)*x^2*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + Sqrt[2]*
b^(1/4)*x^2*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]] + Sqrt[2]
*b^(1/4)*x^2*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]])/(16*a^(
5/4)*x^2)

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Maple [A]  time = 0.007, size = 144, normalized size = 0.7 \[ -{\frac{\sqrt{2}}{16\,a}\ln \left ({1 \left ({x}^{4}-\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{4}+\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{8\,a}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{8\,a}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{1}{2\,a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^8+a),x)

[Out]

-1/16/a/(a/b)^(1/4)*2^(1/2)*ln((x^4-(a/b)^(1/4)*x^2*2^(1/2)+(a/b)^(1/2))/(x^4+(a
/b)^(1/4)*x^2*2^(1/2)+(a/b)^(1/2)))-1/8/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/
b)^(1/4)*x^2+1)-1/8/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^2-1)-1/2/
a/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233486, size = 186, normalized size = 0.92 \[ -\frac{4 \, a x^{2} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}}}{b x^{2} + b \sqrt{\frac{b x^{4} - a^{3} \sqrt{-\frac{b}{a^{5}}}}{b}}}\right ) + a x^{2} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}} + b x^{2}\right ) - a x^{2} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}} + b x^{2}\right ) + 4}{8 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^3),x, algorithm="fricas")

[Out]

-1/8*(4*a*x^2*(-b/a^5)^(1/4)*arctan(a^4*(-b/a^5)^(3/4)/(b*x^2 + b*sqrt((b*x^4 -
a^3*sqrt(-b/a^5))/b))) + a*x^2*(-b/a^5)^(1/4)*log(a^4*(-b/a^5)^(3/4) + b*x^2) -
a*x^2*(-b/a^5)^(1/4)*log(-a^4*(-b/a^5)^(3/4) + b*x^2) + 4)/(a*x^2)

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Sympy [A]  time = 1.99331, size = 34, normalized size = 0.17 \[ \operatorname{RootSum}{\left (4096 t^{4} a^{5} + b, \left ( t \mapsto t \log{\left (- \frac{512 t^{3} a^{4}}{b} + x^{2} \right )} \right )\right )} - \frac{1}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**8+a),x)

[Out]

RootSum(4096*_t**4*a**5 + b, Lambda(_t, _t*log(-512*_t**3*a**4/b + x**2))) - 1/(
2*a*x**2)

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GIAC/XCAS [A]  time = 0.231987, size = 263, normalized size = 1.3 \[ -\frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2}} - \frac{1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^3),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2)*(a/b)^(1/4))/
(a/b)^(1/4))/a^2 - 1/8*sqrt(2)*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2)*(2*x^2 - sqr
t(2)*(a/b)^(1/4))/(a/b)^(1/4))/a^2 - 1/16*sqrt(2)*(a*b^3)^(1/4)*x^4*ln(x^4 + sqr
t(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/a^2 + 1/16*sqrt(2)*(a*b^3)^(1/4)*x^4*ln(x^4 -
sqrt(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/a^2 - 1/2/(a*x^2)